Cluster duality and mirror symmetry for Grassmannians
Cluster duality and mirror symmetry for Grassmannians
复制标题
格拉斯曼的簇对偶性和镜像对称性
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发表时间：
2015
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通讯作者：
L. Williams
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文献类型：
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作者：
K. Rietsch;L. Williams
In this article we use the cluster structure on the Grassmannian and the combinatorics of plabic graphs to exhibit a new aspect of mirror symmetry for Grassmannians in terms of polytopes. For our $A$-model, we consider the Grassmannian $mathbb X=Gr_{n-k}(mathbb{C}^n)$. The $B$-model is a Landau-Ginzburg model $(check{mathbb X}^circ, W_q:check{mathbb X}^circ o mathbb{C})$, where $check{mathbb X}^circ$ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian $check{mathbb X} = Gr_k((mathbb{C}^n)^*)$, and the superpotential $W_q$ has a simple expression in terms of Plucker coordinates, see [MarshRietsch]. From a given plabic graph $G$ we obtain two coordinate systems: using work of Postnikov and Talaska we have a positive chart $Phi_G:(mathbb{C}^*)^{k(n-k)} o mathbb X$ in our $A$-model, and using work of Scott we have a cluster chart $Phi_G^{vee}:(mathbb{C}^*)^{k(n-k)} o check{mathbb X}$ in our $B$-model. To each positive chart $Phi_G$ and choice of positive integer $r$, we associate a polytope $NO_G^r$, which we construct as the convex hull of a set of integer lattice points. This polytope is an example of a Newton-Okounkov polytope associated to the line bundle $mathcal O(r)$ on $mathbb X$. On the other hand, using the cluster chart $Phi_G^{vee}$ and the same positive integer $r$, we obtain a polytope $Q_G^r$ -- described in terms of inequalities -- by "tropicalizing" the composition $W_{t^r}circ Phi_G^{vee}$. Our main result is that the polytopes $NO_G^r$ and $Q_G^r$ coincide.
In this article we use the cluster structure on the Grassmannian and the combinatorics of plabic graphs to exhibit a new aspect of mirror symmetry for Grassmannians in terms of polytopes. For our $A$-model, we consider the Grassmannian $mathbb X=Gr_{n-k}(mathbb{C}^n)$. The $B$-model is a Landau-Ginzburg model $(check{mathbb X}^circ, W_q:check{mathbb X}^circ o mathbb{C})$, where $check{mathbb X}^circ$ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian $check{mathbb X} = Gr_k((mathbb{C}^n)^*)$, and the superpotential $W_q$ has a simple expression in terms of Plucker coordinates, see [MarshRietsch]. From a given plabic graph $G$ we obtain two coordinate systems: using work of Postnikov and Talaska we have a positive chart $Phi_G:(mathbb{C}^*)^{k(n-k)} o mathbb X$ in our $A$-model, and using work of Scott we have a cluster chart $Phi_G^{vee}:(mathbb{C}^*)^{k(n-k)} o check{mathbb X}$ in our $B$-model. To each positive chart $Phi_G$ and choice of positive integer $r$, we associate a polytope $NO_G^r$, which we construct as the convex hull of a set of integer lattice points. This polytope is an example of a Newton-Okounkov polytope associated to the line bundle $mathcal O(r)$ on $mathbb X$. On the other hand, using the cluster chart $Phi_G^{vee}$ and the same positive integer $r$, we obtain a polytope $Q_G^r$ -- described in terms of inequalities -- by "tropicalizing" the composition $W_{t^r}circ Phi_G^{vee}$. Our main result is that the polytopes $NO_G^r$ and $Q_G^r$ coincide.